Fabricating enclosed shapes of multiple widths (from narrow bands to longer tubes) from stock of various shapes such as flat, round, triangular, square, domed etc. of a particular thickness, to achieve a specific internal dimension, diameter, or particular ring fit, is a crucial task for many metal smiths and other fabricators. For instance, consider the case of jewelers who fabricate various enclosed shapes such as rings and bezels from stock. If an enclosed shape of an incorrect internal diameter results from forming an enclosed shape out of a length of stock, then rectifying this mistake is a time and resource intensive process, thus increasing fabrication time and costs.
Given the variability in thicknesses of material, determining the appropriate length of stock required to form an enclosed shape such as a ring with a specific internal diameter, or particular fit, is a complex problem, one which is currently solved through mathematical calculations, charts, guesswork or some combination of these approaches. One sphere in which this problem is particularly vivid is that of jewelers and metal smiths, who fabricate items with variable widths and close tolerances such as rings and bezels for setting stones. The correct sizing of a ring is crucial for comfortable wearing of such an item. Currently, for jewelers fabricating rings out of various types of flat or wire metal stock, they generally rely on a paper chart (size table) to determine the length of stock required for a ring of a specific size (internal diameter). One axis of the size table will indicate specific ring sizes, another axis particular gauges or thicknesses of metal. However this method is inadequate for two reasons Firstly, the width of the finished ring affects the internal diameter required to achieve the desired fit of the ring. Secondly, such tables are hard to use, as they do not directly interact with the material to be measured. This second difficulty with the table method can be illustrated with those learning the craft of metal fabrication, a significant subset of users of such tables. Many beginning jewelers and metal smiths are unaware of the correct way to measure stock using a ruler. Further complicating the issue is the fact that stock measurements for specific ring sizes are expressed in millimeters, which is necessary due to the small sizes at work. However in the US, many individuals, including many beginning jewelers and metal smiths are unfamiliar with this unit of measurement, further complicating the use of a table. A final difficulty is that reading the table itself can be confusing. Such tables are densely packed rows and columns of numbers that are only fractionally different in certain cases, and it is easy to mistake one size for another. The practical difficulties that such problems cause can be highlighted by noting that there is very little tolerance in the ratio between length of stock and ring size when you work on a small scale, and even minor variations in measurement can result in significant shifts in the internal diameter and fit of the ring.
There is another type of printed chart that suffers from different difficulties. These printed charts contain depictions of the dimensions of the length of stock required to form an enclosed shape with a specific internal diameter. These depictions are in the form of various printed rulers for various gauges or thicknesses of stock, or sometime rectangles with size indications. Such depictions also contain an inherent flaw, which is that printing such charts can alter the dimensions of the forms on the chart (scaling issues), thus making accurate measurement impossible. In addition to the difficulty of scaling issues, depiction charts, like tables, do not directly interact with the stock to be fabricated thus not directly aiding the measuring process.
Once an enclosed shape such as a ring has been formed, the accepted method for determining the size or internal diameter of a ring is to place it on a mandrel. A mandrel is a gradually tapered cone of metal with a determinate shape in cross section used to measure or form enclosed shapes. Ring mandrels are typically round, ovoid, or other close variations, and are either forming mandrels with smooth surfaces and may or may not have markings, or measuring mandrels that have ring sizes or internal diameter measurements noted down one side. You measure the internal dimensions or size of an enclosed shape such as a ring by sliding it onto the mandrel and noting the size at the point that it reached. Thus you cannot accurately measure the size or internal diameter of the ring until it has been joined and shaped, a join that it will take significant time and resource to rectify if your initial size is incorrect. If your initial ring is too small, and if the difference is minor, you may be able to stretch it, but this is likely to mar the metal requiring greater effort in finishing. Otherwise it must be cut open, and more material added (by mathematical calculations, charts, guesswork or some combination of these approaches), and the ring re-soldered, creating a ring with two seams rather than one, both compromising the integrity of the ring and increasing finishing time. If the ring is too large, then it must be cut open, excess material removed (again by mathematical calculations, charts, guesswork or some combination of these approaches), and re-soldered, again requiring extra finishing. Moreover if the first rectification does not work, then this may happen again. Note that this problem will occur no matter what type of enclosed shape is formed through taking a length of stock, joining it into an enclosed shape, and then forming it over an appropriately shaped mandrel. For instance bezels and other components that come in a wide variety of enclosed shapes including squares, octagons, triangles, rectangles etc. These types of mistakes add substantial time and production cost to the fabrication process, while possibly compromising the look and integrity of the finished product.
In short the problem is that there are in fact three variables that determine the internal diameter and particular fit of an enclosed shape such as a ring. These variables are: (1) the length of the stock; (2) the thickness (gauge) of the stock; and (3) the width of the proposed enclosed shape. Fingers are not of a uniform diameter down the length of a finger, so a wide ring, which covers more of the finger, will fit differently to a narrow ring, which is only concerned with the diameter of the finger at a specific point. Consequently, a wide ring and a narrow ring of the same size will require different lengths of stock of the same gauge to form. This variability is not taken into account with the current methods of providing length to gauge (thickness) ratios. A further inconvenience is generated by the fact that most metal smiths work in multiple thicknesses (gauges) of metal, where differences in the thickness of the stock is so small as to be difficult to perceive by eye, but still significantly affects the internal diameter of the finished enclosed shape. This means that mistaking one gauge of metal for another can further complicate the fabrication process.